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Which of the following trigonometric identities are true? \(\begin{aligned} & \text{I. } \sin ^2\left(41^{\circ}\right)+\sin ^2\left(49^{\circ}\right)=1 \\ & \text{II. } \sin…

Question and options

Which of the following trigonometric identities are true?

I. sin⁡2(41∘)+sin⁡2(49∘)=1II. sin⁡2(60∘)−2tan⁡(45∘)−cos⁡2(30∘)=−1III. sin⁡2(θ)+11+tan⁡2(θ)=1\begin{aligned} & \text{I. } \sin ^2\left(41^{\circ}\right)+\sin ^2\left(49^{\circ}\right)=1 \\ & \text{II. } \sin ^2\left(60^{\circ}\right)-2 \tan \left(45^{\circ}\right)-\cos ^2\left(30^{\circ}\right)=-1 \\ & \text{III. } \sin ^2(\theta)+\frac{1}{1+\tan ^2(\theta)}=1 \end{aligned}
  1. I and II only
  2. I and III only
  3. II and III only
  4. I, II and III

Answer and explanation

Correct answer: B - I and III only
I: sin²41° + sin²49° = sin²41° + cos²41° = 1, so it is true. II: 3/4 − 2(1) − 3/4 = −2, not −1, so it is false. III: 1/(1 + tan²θ) = cos²θ, so sin²θ + cos²θ = 1 is true. Hence I and III only.

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